If a Muggle-born Witch or Wizard was to die and they have a Gringott's account, the contents of which have been left to a non-magical Muggle within their family, how do Gringott's contact that person to inform them of their inheritance? Or would there be a Ministry of Magic law against such a thing occurring in order to protect the magical world from exposure to Muggles?
The probate system is more complex than you're assuming. Gringott's wouldn't necessarily have to contact directly the person to whom the assets were left.
The deceased Muggle-born witch or wizard, when writing their will, should have named an executor, whose role is as follows (emphasis mine):
a person named by the maker of a will or nominated by the testator to carry out the instructions of the will. Typically, the executor is the person responsible for offering the will for probate, although it is not required that he fulfill this. The executor's duties also include disbursing property to the beneficiaries as designated in the will, obtaining information of potential heirs, collecting and arranging for payment of debts of the estate and approving or disapproving creditors' claims. An executor will make sure estate taxes are calculated, necessary forms are filed, and tax payments are made. They will also assist the attorney with the estate. Additionally, the executor acts as a legal conveyor who designates where the donations will be sent using the information left in bequests, whether they be sent to charity or other organizations. In most circumstances, the executor is the representative of the estate for all purposes, and has the ability to sue or be sued on behalf of the estate. The executor holds legal title to the estate property, but may not use the title or property for his/her own benefit, unless permitted by the terms of the will.
Being aware of the International Statue of Wizarding Secrecy, they would naturally make sure that the executor is either a wizard or a Muggle close enough to the wizarding world to know about it (e.g. their spouse or parent). They would not, of course, be restricted in their choice of beneficiaries: they can leave money or other assets to whomever they wish, including Muggles. It would be up to the executor to ensure that the Statute of Secrecy is not broken, e.g. by withdrawing gold from Gringott's and converting it to Muggle money before distributing it to Muggle beneficiaries.